Hello and congratulations on the new institute "Discrete Structures and Symbolic Computing"! The name sounds exciting, but what exactly is behind it?
Frederik Witt: Discrete structures or discrete mathematics form a subfield of mathematics, which originally dealt with mathematical operations on finite or at most countably infinite sets such as the natural numbers. Classical subfields are for example combinatorics or number theory. In the meantime, discrete mathematics has established close relations to algebra, logic and theoretical computer science, but discrete methods are also successfully applied in areas such as dynamical systems or differential geometry.
On the other hand, symbolic computation includes computer-based methods whose focus is on finding exact solutions using algebraic algorithms. In this sense, symbolic computation can be seen as complementary to scientific computation, which is rather based on numerical and approximate-analytical methods. Application areas are all problems that can be described by polynomial equations, e.g. classical questions from discrete optimization like the backpack problem: A thief wants to carry away objects of different volume and value in his backpack with a given volume capacity. How then does he find the optimal value while respecting the backpack volume?
And the logo?
Meinolf Geck: Discrete structures also play a major role in Lie theory. These include the so-called root systems, which are extremely symmetric configurations of vectors that were already described by Killing at the end of the 19th century. The logo is the 2-dimensional projection of the E8 root system in 8-dimensional space. The image itself is by John Stembridge (University of Michigan).
We know from previous interviews with you that your respective fields of work are algebra and geometry, classic areas of pure mathematics. How does that fit with computers?
Frederik Witt: One must not forget that applications have often arisen from theoretical problems -- just think of quantum mechanics or computer science! Now, for geometric reasons, the question of invariant polynomials under certain group effects is important, a question that founded the so-called invariant theory with early protagonists like Hilbert and eventually led to Mumford's geometric invariant theory or Hironaka's theory for the resolution of singularities, both results that were rewarded with the Fields Medal. It is not without reason that SINGULAR, which was developed at the University of Kaiserslautern, is also the name of one of the leading software packages in algebraic geometry.
Meinolf Geck: Besides, this also illustrates very nicely the intrinsic connection between groups and geometry! Moreover, there is a long tradition of computer use in group theory, especially in connection with the classification of finite simple groups, one of the most important results of 20th century mathematics ever. To this day, the existence of some "sporadic" simple groups can only be shown with computer support, and computers are also essential in the further study of simple groups.
And how did you personally come to symbolic computation?
Meinolf Geck: After my studies, I was an assistant at the Chair D for Mathematics at the RWTH Aachen University for many years, where the first versions of the program system GAP were developed at that time -- by the way, originally from four diploma theses. I was not directly involved in the development, but I became one of the first users, so to speak, and immediately wrote a program package for calculating with root systems and Weyl groups. This was enormously useful, because with such programs one can test working hypotheses in a matter of seconds and in fortunate cases also get new ideas. Over time, this package has been absorbed into the much broader CHEVIE project. Being able to systematically experiment with hypotheses and substantive examples in a system like GAP has been and continues to be an essential foundation for a number of my papers.
Frederik Witt: I also started my studies in Aachen, where I came into contact with MAPLE at an early stage -- it was not as common then as it is now. Later, in my PhD, I intensively used LiE, a software package for Lie groups. As a PostDoc, I finally started to work more intensively with computer algebra and especially with SINGULAR.
How did the founding of the institute come about?
Frederik Witt: Our common hook was initially teaching. To illustrate abstract theory, many examples are simply essential, and these can easily be generated with the computer. In addition, programming helps to clearly differentiate mathematical concepts. For example, k[x] can be a ring, a k-vector space, or a k-algebra, depending on the context. But, of course, this has to be told to the computer, which helps to strictly distinguish these conceptualizations. Both aspects are related and support each other. We then developed the GAGA profile line in order to visibly map the already existing activities in teaching as well.
Meinolf Geck: As far as research is concerned, there is a rich tradition of symbolic computation and the development of algorithmic methods in algebra and geometry, especially in Germany. This has been supported by the DFG for decades through collaborative programs, in which the University of Stuttgart has also regularly participated. Currently, we are project leaders or members in a 12-year SFB-TRR "Symbolic Tools", in which several disciplines (algebra, geometry, number theory, free probabilities, ...) interact through the common bracket of "algorithms", which is thematically highly interesting. Due to this angfristige engagement and the combined interests of our working groups, it seemed to make sense to raise our profile also outwardly by an own institute. And we hope that this will also benefit the mathematics department as a whole.
Frederik Witt Certainly our common background also helped. We had some of the same academic teachers in Aachen, and we are also connected by our many years of experience abroad.
What projects do you want to tackle in the near future?
Meinolf Geck: It will be interesting to see how program systems like GAP, MAPLE, etc. develop. Within the framework of the above-mentioned SFB-TRR, a new such system is being developed on the basis of the relatively young programming language Julia (which was originally conceived by applied mathematicians at MIT). This opens up perspectives for new fields of application of computer-aided methods in pure mathematics, and in particular hopefully in the areas that interest me.
Frederik Witt: I also expect exciting prospects for my field of geometry, for example in toric geometry, a particularly combinatorial area of algebraic geometry. And of course we hope that the new institute will become a natural docking station for further projects and collaborations.
Thank you very much for the interview.
Prof. Frederik Witt
Prof. Meinolf Geck
Institute for Discrete Structures and Symbolic Computation (IDSC)
